Abstract

Let , be the -Bernstein polynomials of a function . It has been known that, in general, the sequence with is not an approximating sequence for , in contrast to the standard case . In this paper, we give the sufficient and necessary condition under which the sequence approximates for any in the case . Based on this condition, we get that if for sufficiently large , then approximates for any . On the other hand, if can approximate for any in the case , then the sequence satisfies .

1. Introduction

Let . For any nonnegative integer , the -integer is defined by
and the -factorial by
For integers , with , the -binomial coefficient is defined by
In [1], Phillips proposed the -Bernstein polynomials: for each positive integer and , the -Bernstein polynomial of is
where
Note that, for , is the classical Bernstein polynomial :

In recent years, the -Bernstein polynomials have been investigated intensively and a great number of interesting results related to the -Bernstein polynomials have been obtained. Reviews of the results on -Bernstein polynomials are given in [2, Chapter 7] and [3, 4].

The -Bernstein polynomials inherit some of the properties of the classical Bernstein polynomials, for example, the end-point interpolation property and the shape-preserving properties in the case , representation via divided differences. We can also define the generalized Bézier curve and de Casteljau algorithm, which can be used for evaluating -Bernstein polynomials iteratively. These properties stipulate the importance of -Bernstein polynomials for the computer-aided geometric design. Like the classical Bernstein polynomials, the -Bernstein polynomials reproduce linear functions and are degree reducing on the set of polynomials. Apart from that, the basic -Bernstein polynomials admit a probabilistic interpretation via the stochastic process and the -binomial distribution in the case ; see [5].

On the other hand, when passing from to convergence properties of the -Bernstein polynomials dramatically change. More specially, in the case , are positive linear operators on , and the convergence properties of the -Bernstein polynomials have been investigated intensively (see, e.g., [6–11]). In the case , are not positive linear operators on , and the lack of positivity makes the investigation of convergence in the case essentially more difficult. There are many unexpected results concerning convergence of -Bernstein polynomials in the case (see [2, 12–17]). For example, the rate of approximation by -Bernstein polynomials in for functions analytic in is versus for the classical Bernstein polynomials, while, for some infinitely differentiable functions on , their sequences of -Bernstein polynomials may be divergent (see [12]). In [2, 15], strong asymptotic estimates for the norm as for fixed and as are obtained. It was shown in [2] that faster than any geometric progression for fixed . This fact provides an explanation for the unpredictable behavior of -Bernstein polynomials () with respect to convergence.

This paper is devoted to studying approximation properties of -Bernstein polynomials for taking varying values that tend to 1. We note that, from the very first papers (see [1]), there was interest in such approximation properties. In the case , many interesting results including the convergence, the rate of convergence, Voronvskaya-type theorems, and the direct and converse theorem are obtained (see [1, 6, 8–11]). It was shown in [1, 8] that, in the case , the condition is necessary and sufficient for the sequence to be approximating for any .

Naturally, the question arises as to whether the sequence to be approximating for any as tends to 1 from above. It turns out that, in general, the answer is negative. Indeed, Ostrovska showed in [13] that if slower than , then the sequence may not be approximating for some (e.g., ). However, in [14] Ostrovska showed that if fast enough, the sequence is approximating for any : a sufficient condition is .

In this paper, we continue to study the convergence of the sequence as tends to 1 from above. Clearly, the convergence of the sequence depends heavily on the operator norms . We remark that for for all . In contrast to this, vary with . By the delicate analysis of , we obtain the sufficient and necessary condition under which approximates for any . Based on this condition we get that if can approximate for any , then the sequence satisfies . On the other hand, if for sufficient large , then approximates for any .

2. Statement of Results

From here on we assume that . The following theorem gives the sufficient and necessary condition for convergence of the sequence for any .

Theorem 1. Let . Then the sequence converges to in for any if and only if

Based on Theorem 1, we obtain the following necessary condition for convergence of the sequence . Indeed, we show that if , then with .

Theorem 2. Let . If the sequence converges to in , for any , then

Finally, we give the sufficient condition for convergence of the sequence .

Theorem 3. Let . If the sequence satisfies for sufficiently large , then, for any , converges to uniformly on .

Corollary 4. Let . If the sequence satisfies
then, for any , converges to uniformly on .

Remark 5. Using the same technique as in the proof of Theorem 3, we can prove a slightly stronger conclusion: if
for some positive constant and sufficiently large , then, for any , converges to uniformly on .

For , we set
Let , . Clearly,
Note that for and for and . This means that
It follows that

Proof of Theorem 1. From Corollary 7 in [12] we know that, for any polynomial , we have
uniformly in as . It follows from the well-known Banach-Steinhaus theorem that approximates for any if and only if
We set
Since for and , we get, for ,
and, therefore,
Next we will show that
Note that, for ,
If we show that, for and ,
then
and (20) follows. Indeed, for and ,
Hence, (22) is equivalent to the following inequality:
which is also equivalent to the inequality
For and , we have
This proves (26). On the other hand, for , which completes the proof of (20). From (14), (19), and (20), we get
This implies that (16) is equivalent to
Theorem 1 is proved.

Proof of Theorem 2. First we show that
Otherwise, we may assume that
which implies
We have
This leads to a contradiction by Theorem 1. Hence, (30) holds.Next, we show Theorem 2. Assume that . Then by (30) we may suppose that, for some , , ,
For , we set , . Direct computation gives that
Since the function is convex on for a fixed , we get that
This means that and is nonincreasing on . Hence, for , , we have
Put . Then, for , we have
Using (34), the inequalities
and the nonincreasing property of , we continue to obtain that
We observe that
and, for , ,
Thus, for some and sufficiently large , we have
By Theorem 1, we know that there exists a function such that the sequence does not converge to in . This leads to a contradiction. Hence, . Theorem 2 is proved.

Proof of Theorem 3. From Theorem 1, we know that it is sufficient to show that if for sufficiently large , then
For , we set . Then and . Since, for ,
by (37) we get that
On the other hand, by (37) we have
It follows from (46) and (47) that
Hence, for , ,
Obviously (49) is satisfied for . We note that, for ,
The above formula with means that
Thus, by (49),
This completes the proof of Theorem 3.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is very grateful to S. Ostrovska and the anonymous referees for their useful comments and suggestions that helped improve the presentation of the paper. The research is supported by the National Natural Science Foundation of China (Project no. 11271263) and the Beijing Natural Science Foundation (1132001).